A tropical version of Nevanlinna theory is described in which the role of meromorphic functions is played by continuous piecewise linear functions of a real variable whose one-sided derivatives are integers at every point. These functions are naturally defined on the max-plus (or tropical) semi-ring. Analogues of the Nevanlinna characteristic, proximity and counting functions are defined and versions of Nevanlinna's first main theorem, the lemma on the logarithmic derivative and Clunie's lemma are proved.As well as providing another example of a tropical or dequantized analogue of an important area of complex analysis, this theory has applications to so-called ultra-discrete equations. Preliminary results are presented suggesting that the existence of finite-order max-plus meromorphic solutions can be considered to be an ultra-discrete analogue of the Painlevé property.